10/13/2009, 03:53 AM
(This post was last modified: 10/13/2009, 08:07 PM by Base-Acid Tetration.)
Now that we have the complex plot of tetration, let's look for the complex fixed points.
I know we have the real fixed point at approx. -1.850354529027181418483437788.
I don't have the proper precise tools to investigate the complex fixed points' existence and location with much detale. I am sorry to overburden you, but if someone posted calculation/complex plot of tet(z)-z it would be very nice, as we could see the fixed points as zeros of the function.
If there are complex fixed points, I think the first pair will be somewhere near -W(-1) (+-conjugate).
There probably are more fixed points at impractically large values of z with some specific ratio of real and imaginary parts. (the repeating tetration-pattern fractal roughly repeats at 3.something +- .something*i; the ratio of the real part vs. imaginary part will be relevant, as well as what the fractal (the large-modulus parts of tet(z)) is that repeats. to look for those large fixed points we may use
tlog(tet(z)) - tlog(z) or, to the extent that tlog and tet cancel out, z - tlog(z) (tlog is tetra-logarithm). then we may tetrate the large zeros of z - tlog(z) to get the true value of the fixed points.
Pentation is depending at us!
I know we have the real fixed point at approx. -1.850354529027181418483437788.
I don't have the proper precise tools to investigate the complex fixed points' existence and location with much detale. I am sorry to overburden you, but if someone posted calculation/complex plot of tet(z)-z it would be very nice, as we could see the fixed points as zeros of the function.
If there are complex fixed points, I think the first pair will be somewhere near -W(-1) (+-conjugate).
There probably are more fixed points at impractically large values of z with some specific ratio of real and imaginary parts. (the repeating tetration-pattern fractal roughly repeats at 3.something +- .something*i; the ratio of the real part vs. imaginary part will be relevant, as well as what the fractal (the large-modulus parts of tet(z)) is that repeats. to look for those large fixed points we may use
tlog(tet(z)) - tlog(z) or, to the extent that tlog and tet cancel out, z - tlog(z) (tlog is tetra-logarithm). then we may tetrate the large zeros of z - tlog(z) to get the true value of the fixed points.
Pentation is depending at us!